Mathematical educational apparatus using blocks



I. R. BARROWS 3,403,460

MATHEMATICAL EDUCATIONAL APPARATUS USING BLOCKS Oct. 1, 1968 '7 Sheets-Sheet 5 Filed July 1966 Ask $1 Oct. 1, 1968 I. R. BARROWS 3,403,450

MATHEMATICAL EDUCATIONAL APPARATUS USING BLOCKS Filed July 5, 1966 '7 Sheets-Sheet 4 F I G. 6. 3 zszsqm z 5 4 K 3| Oct 1968 I. R. BARROWS 3,403,460

MATHEMATICAL EDUCATIONAL APPARATUS USING BLOCKS 7 Sheets-Sheet 5 Filed July 5, 1966 United States Patent 3,403,460 MATHEMATICAL EDUCATIONAL APPARATUS USING BLOCKS Irvin R. Barrows, 1616 S. Compton St., St. Louis, Mo. 63104 Filed July 5, 1966, Ser. No. 562,867 Claims. (CI. 35-31) ABSTRACT OF THE DISCLOSURE Educational apparatus particularly suitable for teaching mathematics is disclosed. The apparatus comprises a tray receiving an answer card and a plurality of blocks. The answer card has indicia thereon forming numbered spaces and the number of blocks equals the number of spaces. Each block is numbered to correspond to one of the spaces and the blocks are coded to form a recognizable pattern when the blocks are properly arranged on the spaces.

This invention relates to educational apparatus particularly suitable for teaching mathematics such as addition, subtraction, multiplication and factoring, for example.

Among the several objects of the invention may be noted the provision of apparatus for teaching mathemati s in which solutions to problems are easily and quickly graded by the teacher; the provision of educational apparatus which permits the teacher to simultaneously and quickly grade a plurality of problems worked by a student instead of grading each problem individually, thereby reducing the time required by the teacher for each student; the provision of apparatus for teaching mathematics wherein answers to problems are graded by reference to coding on elements of the apparatus rather than by direct inspection of the answer, and to the provision of such apparatus wherein the coding for the solution to several problems are related to permit grading of the several problems at once. Other objects and features will be in part apparent and in part pointed out hereinafter.

The invention accordingly comprises the constructions hereinafter described, the scope of the invention being indicated in the following claims.

In the accompanying drawings, in which several of various possible embodiments of the invention are illustrated,

FIG. 1 is an exploded perspective showing one embodiment of educational apparatus of this invention;

FIG. 2 is a plan view of the card of the FIG. 1 apparatus;

FIG. 3 is a perspective of one of the blocks of FIG. 1;

FIG. 4 is a plan showing a pattern formed by the blocks of the FIG. 1 apparatus;

FIGS. 5 and 6 are views showing how the FIG. 4 pattern Was formed;

FIG. 7 is a view illustrating another method for forming a pattern for the blocks;

FIG. 8 is a fragmentary view showing the FIG. 4 pattern after working addition problems;

FIG. 9 is a view similar to FIG. 5 showing the solution to subtraction problems;

FIG. 10 is a view of an answer card used for teaching multiplication;

FIG. 11 is a view showing for teaching multiplication;

FIG. 12 is a view showing a factoring answer card;

FIG. 13 is a view illustrating apparatus for a base four system using colored edge coding for grading a problem;

FIG. 14 is a view similar to FIG. 13 showing the use of shaped block edges for coding; and

FIG. 15 is a view showing different shaped edges for the pattern on blocks used ice answer card spaces especially useful in teaching proper hand motion.

Corresponding reference characters indicate corresponding parts throughout the several views of the drawings.

Referring now to FIG. 1 of the drawings, educational apparatus of the invention comprises a tray 1 which is adapted to receive an answer card 3 and a plurality of blocks 5. The center portion of tray 1 is recessed as shown at 7, the recess being of sufficient size and shape to receive the answer card 3 and the blocks 5. Edge or border portions 9 of the tray limit lateral movement of the answer card and blocks 5 when they are positioned on the tray.

The answer card 3 as shown in FIGS. 1 and 2 is a flat, thin piece of stiff paper, cardboard or the like having flat upper and lower surfaces. The upper surface of card 3 is divided by a series of lines 11 into ten rows designated R1 through R10 extending transversely entirely across the surface of the card from one side edge to the other side edge. The same surface of the card is divided into ten columns designated C1 through C10 by a plurality of spaced, generally parallel lines 13 which run from the upper edge to the lower edge of the card. Lines 11 and 13 form a plurality of square boxes or spaces 15 each of which is the same size and shape. There are 10 spaces 15 in each row and there are 10 columns of the rows. Each space 15 bears numerical indicia designated 17. The particular indicia chosen depends upon the use of the apparatus. As illustrated in FIG. 2, the numerical indicia 17 comprise the numbers 1 through with the numerals being arranged in columns of 10 each beginning with the number 1 in the upper row and left column and extending down the rows of the left column, then to the top row of the next column and consecutively through that column, etc. It will be understood that the numbers can be arranged in other ways, such as arranging the numbers 1-10 entirely in the row R1 and consecutively numbered from column C1 through column C10, then arranging the numbers 11-20 in row R2 immediately beneath the first 10 numbers, etc. When the apparatus is used for teaching addition or subtraction, the spaces in a column (or row) should be consecutively numbered and the space numbers in the last row (R10) should be numbered one less than the space numbers in the next column of the first row. No space or block should be separate from the others. Each space may contain a second number (not shown) of a different color which will be helpful to the pupil in working addition and subtraction problems. For example, the spaces numbered 1 to 50 can also be numbered 101 to 150, respectively, and the spaces numbered 51 to 100 can also be numbered 49 to 0, respectively. If the apparatus is to be used for teaching only one type of problem, then the indicia on card 3 can be placed directly on tray 1 and the card can then be eliminated.

One of the blocks 5 is illustrated in detail in FIG. 3 of the drawings and is shown to comprise a first face 5a having an opposite face 5b, a third face 50 and an opposite face 5d, a fifth face 52 and its opposite face 5). Each of the faces Sa-Sf is preferably the same size and shape and each is preferably substantially the same size and shape as the spaces 15 on the answer card 3. There is one block for each of the spaces 15 on the answer card. Face 5a contains numerical indicia 19 and the numerical indicia 19 on the block correspond to one of the numbers or indicia 17 on the answer card. The numeral 1 is shown in FIG. 3. Thus for the 100 spaces 15 illustrated in FIG. 2 100 blocks 5 are provided each of which has a number on its face 5a corresponding to only one of the numbers on the answer card and being different from each of the numbers 19 on the other blocks. In other words for the space numbered 1 on the answer card there is only one block numbered 1.

When working problems (as explained later), the blocks are placed on the spaces of the answer card 3 so that the numerical indicium 19 on any particular block has a specific mathematical relation to the numerical indicium 17 on the particular space on the card occupied by that block. When the problems have been correctly worked, this same mathematical relation exists between every block and the indicia on the space it occupies. In other words, if the number 1 block is on the number 1 space, the number 2 block is on the number 2 space, the number 3 block is on the number 3 space, etc., then the numerical relation between the indicia on the blocks and the space is a 1-1 ratio. On the other hand, the blocks can be placed on the spaces so that the number 1 block is on the space marked number 2, the number 2 block is on the space marked number 3, the number 100 block is on the space 15 marked 1 (and which may also be marked 101, etc. Then the numerical indicia on the blocks would be one less than the numerical indicia on the spaces.

FIG. 4 of the drawings illustrates all of the blocks 5 arranged in rows and columns as they would appear if the blocks were positioned on the answer card 3 with the numbered faces 5:: of each block facing downwardly and resting on the space on the answer card containing the same number as the indicia on the block. In other words, the number 1 block rests on the number 1 space, the number 100 block rests on the number 100 space, etc. As shown in FIG. 4, the faces 5b of the blocks face upwardly and each contains coding. The coding illustrated on each block in FIG. 4 comprises indicia forming a portion of a pattern which (as a whole) is generally designated 21. The pattern indicia on each block is related to the pattern indicia on all of the other blocks and to the numerical indicia on the blocks so that the indicia on the individual blocks jointly form an unbroken distinctively recognizable pattern when the blocks are arranged with their faces 5b facing upwardly and with their faces 5a containing the numerical indicia resting on the space bearing the same number on the card 3. The pattern 21 is selected so that each block 5 may be moved to other positions on the answer card, while still maintaining a given numerical relation between the indicia on each block and the indicia on the space it occupies, without breaking the distinctively recognizable pattern 21 even though it may cause a shift in the pattern as will be more apparent from the following description.

The pattern indicia on each of the blocks are distinctive and unique, that is, no two blocks bear exactly the same pattern indicia. Thus the pattern 21 will become broken or irregular if any block is improperly placed on the answer card so that its numerical indicia does not have the same mathematical relation to the indicia on the card space occupied by it as the corresponding indicia on all the other blocks have to the card spaces occupied by them.

The pattern shown in FIG. 4 is merely illustrative of various patterns which can be used on the blocks. The pattern 21 comprises stripes of three different colors and of various widths extending diagonally across the faces 51) of the blocks. The diagonal stripes extending from the upper left to the lower right in FIG. 4 comprise relatively wide stripes 23a and relatively narrow stripes 23b of the same color, such as orange, relatively wide stripes 24a and relatively narrow stripes 24b of a second color, such as blue, and relatively wide stripes 25a and relatively narrow stripes 25b of the third color, such as purple. It will be observed that the arrangement or pattern of the colored stripes is a repeating one. Thus the stripe 23a which begins the pattern at the lower left corner of FIG. 4 (the block in column C1, row R10) is the same width and color as the wide stripe 23a which extends from the upper left of the pattern in FIG. 4 (the blocks in row R1, columns C1 and 2) to the lower right (the block in column C10, row R10).

There is also a pattern of lines extending diagonally from the lower left to the upper right..In the pattern 21 these lines or stripes comprise blue stripes 27, orange stripes 29 and purple stripes 31. This pattern repeats itself beginning with the blue stripe 27 extending across the block in column C1, row R1 to the blue stripe running through the block in column C10, row R10.

it least one of the blocks 5 of each set of blocks preferably contains some key indicia or pattern formation so that it can be readily distinguished from others of the blocks. In the pattern shown in FIG. 4 the block in column 1, row 1 is marked with a dot designated 32. This permits the grader or teacher to determine movement of that particular block when shown the solution to a series of problems. The position of this one block, together with the fact that a readily distinguishable unique pattern is formed on the block faces 5b and the fact that each block contains a pattern portion different from the pattern portion on the other blocks, permits the teacher immediately to determine whether the problems given have or have not been correctly solved. Use of the apparatus for solving addition and subtraction problems will now be described.

Assume initially that the pupil has been given the set of problems of adding one to the number on face 5a of each block. Each of the blocks 5 is first placed on a suitable supporting surface with the face Sa facing upwardly so that the numerical indicia 19 on the block faces are visible to the pupil. As an alternative, the blocks may be placed in a suitable container and withdrawn at random one at a time as the problems are worked to insure a random sequence in working the problems. The pupil picks up each block, then adds 1 to the number shown on face 5a of each block and, with lateral rotation (as one turns the pages of a book), he turns each block over and places it on the square of the answer card 3 marked with the answer to the problem. For example, assuming that the dot 32 is on the block 5 having the indicia 1 on it, then when solving the above noted problem of adding one to the number on the block, the pupil looks at the number 1 on the block, adds 1 to it and correctly determines that the answer is two. The pupil then takes the block bearing the indicia 1, turns it over and places it face down on the square marked 2 on the answer card. This is the position shown for this particular block in FIG. 8 of the drawings. In a similar manner the other 99 blocks are placed with the faces 5a down on the answer card so that the number on the answer card occupied by each block is one more than the number on the block. In this manner, a specific mathematical relation is established between any block and the number occupied by it which is the same as the mathematical relation between each of the other blocks and the indicia on the space on the answer card occupied by them. Because of the particular pattern on the faces 5b of the blocks, the problems when correctly solved result in a still readily recognizable pattern appearing on the faces of the blocks when taken as a whole. This pattern is shown in FIG. 8. The pattern is unbroken and is substantially the same as the pattern in FIG. 4 except that it has been shifted as a result of solving the particular set of problems. With the exception of the blocks occupying the row R10 position in FIG. 4, each of the blocks has been shifted downwardly 1 space on the answer card during solution of the problem. As to the blocks occupying the row R10, columns C1-C9 position in FIG. 4, these have shifted to columns C2-C10, respectively in row 1 of FIG. 8. The block which originally occupied the position in column C10, row R10 in FIG. 4, is now at column C1, row R1 in FIG. 8. In order to grade the problems, all the teacher has to do is locate the block having the dot 32 to determine that it has moved to the correct position for the particular set of problems assigned and to quickly glance at the pattern formed by the other blocks to determine that it is regular and uninterrupted. If these two conditions are met, then the pupil has correctly solved the set or prob lems. If just one block is positioned so that the pattern is broken, then there is at least one incorrect solution for the one hundred addition problems worked by the pupil. By using an answer card 3 with numbers from 49 to and from +100 to +150 in spaces 15 (in addition to the numbers designated 17) one can work fifty different sets of one hundred addition problems and fifty different sets of one hundred subtraction problems for a total of ten thousand problems.

In solving a problem a uniform hand motion is required for turning each block over. For example, each block should be rotated or turned laterally 180 to the left as it is moved from the position with its face a facing upwardly to the position where its face 5b faces upwardlynNonuniform rotation of the blocks or rotation in a vertical direction (for the particular pattern and the number indicia shown) will result in a broken pattern even if the problems have been correctly solved.

FIG. 9 illustrates the correct solution to the set of problems of subtracting the number one from the number shown on each block. Note that the block bearing thedistinctive indicia 32 has moved from the row R1, column C1 position to the row R10, column C position. The blocks in row R1, columns C2-C10 in FIG. 4 are located at row R10, columns C1-C9 in FIG. 9. The other blocks have moved up one space from their FIG. 4- positions. Here again the pattern 21 on the blocks is continuous, unique and is uninterrupted so that the teacher can tell at a glance that the set of problems has been worked correctly.

While the particular number of squares on the answer card and the number of blocks used can be varied as desired, it is preferred that the number of squares and blocks be such that each block used can occupy one square during the solution of any addition and subtraction problem. By using 100 blocks and squares as shown in the drawings, the block originally occupying the number 1 spot bearing the indicia 1 on it can be moved during solution ofa subtract 1 problem to the column C10, row R10 position which is either a 100 or a 0 square. This occurs in a base 10 system by using multiples of 100 blocks. For any given base, each space can be assigned by any value which is a multiple of the number of spaces on card 3 plus the number assigned to that space.

The number of blocks used may be different than the number of squares. For example, from 99 to as few as two blocks might be used with the answer card illustrated in FIG. 2. When fewer than 100 blocks are used the indicia and pattern coding on each block should be such that the blocks are adjacent to other blocks on the answer card when problems are properly worked. This permits grading of the solution to problems by reference to the pattern coding only.

When fewer blocks are used than the number of spaces on answer card 3, a special answer card may be used which leaves blank the spaces which are not to be used, or such spaces may be used to tell the student what operation is required for that set of problems. Other forms of answer card 3 may include cards with spaces 15 with no indicia or cards with number indicia 17 superimposed on the same pattern indicia 21 that will be formed when all the blocks 5 have been correctly placed on answer card 3. This will permit pupils to determine the correctness of the solution to the problem as the blocks are placed on the answer card.

The answer card 3 instead of consisting of only straight lines 11 and 13 may consist of straight lines 13 and curved, generally horizontal lines 11a comprising an arch curving upward at the top and bottom of each space 15. This is shown in FIG. 15 for a base four system. Each arch is identical and when blocks 5 are so curved it allows pupils to more quickly learn proper hand motion for placing blocks 15 on the answer card.

FIGS. 5 and 6 illustrate preparation of the pattern 21. One hundred squares are marked off on a sheet of paper with each square representing one of the blocks 5 and preferably being the same size and shape as the blocks. Then the sheet is cut along a diagonal line which is at the same angle as the stripes extending in one direction across the blocks. This out line has been designated 34 in FIGS. 4 and 5. Then the lower-right portion of this sheet designated 36 is placed so that its right edge lies along the left edge of the upper left portion of the sheet and the upper right corner of the lower right portion is positioned adjacent to the upper left corner of the upper left portion of the sheet. Line 34 and the bottom line of the square are then extended downwardly and to the left until they meet, and the small triangle 38 so formed is then moved ten spaces to the left to form the parallelogram shown in FIG. 5.

After the stripes shown in FIG. 5 have been drawn, then the parts of the sheets are returned to their original positions to form a square and line 33 (FIGS. 4 and 6) is drawn. The portion of the sheet which constitutes the upper right part (above line 33) is moved so that the right hand edge is along the left edge of the other portion of the sheet and another parallelogram is formed as shown in FIG. 6. With the portions of the sheet in this position the lines 2311 and 23b, 24a and 24b, and 25a and 25b are drawn. In the particular pattern illustrated the narrower stripes 23b, 24b and 25b are of equal width and constitute approximately /3 of the total width of the pattern at its widest point as viewed in FIG. 5. The wider stripes 23a, 24a and 25a are twice as wide as the narrower stripes and occupy approximately /3 of the total width of the pattern. Then the portions of the sheet are returned to their original position. The resulting pattern on each square on the sheet is then transferred individually to the corresponding block to produce the pattern 21 shown in FIG. 4.

FIG. 7 illustrates another way for laying out a pattern for the blocks. First, a plurality of squares 35 are laid out next to each other. Each square 35 is subdivided into the same number of spaces as the blocks for a given set of blocks, such as into 10 rows and 10 columns to form spaces representing the set of blocks shown in FIGS. 1 and 4. Each square 35 of a lower row of squares is staggered one space to the left of the square 35 above it. Then beginning at any arbitrarily selected point on a space on any one square 35 a line 37 is drawn to a corresponding spot on another one of the squares. The line shown in FIG. 7 is drawn from the block in column 1, row 3 of the upper left square 35 to the corresponding block on the lower right square 35. It will be observed that this line crosses five squares 35 at different parts of the squares. The portions of this line lying in the various squares have been designated 37a, 37b, 37c, 37d and 376. Using the upper left square 35 as the primary square on which the pattern is to be constructed, the portions of the line 37b-37e are transferred onto this upper left square to occupy the same relative position on the upper left square as they occupy on the square they cross. For example, the portion 37b of the line extends across the block in column C1, row R10 in the second square 35 and a corresponding line is drawn on the upper left square and has been designated 37b. Similarly the line portions 37c, 37d and 37e are transposed to the upper left square where they are designated 37c, 37d and 37e'. The line 37 may be given any desired width or color.

Then other lines (not shown) are drawn either at the same or at different angles and the lines may extend from upper left to lower right as viewed in FIG. 7 or from the lower left to the upper right as well as in other directions. Each of the segments of these lines is transposed to the upper left block. A sufficient number of lines, line widths, and colors are used to produce a unique easily recognized pattern which does not become irregular when certain blocks or rows of blocks are shifted relative to other blocks during addition, subtraction, etc. It will be under- I stood .that pattern lines can also be drawn for rectangles which are not base times base (i.e. 10 x 10) size if the above rules are followed for the placing of the rectangles relative to each other.

FIGS. 10. and 11 illustrate educational apparatus of the invention particularly suitable for teaching multiplication. In FIG. 10, an answer card, generally designated 41, is shown to comprise squares or spaced arranged in 10 columns C1-C10 and 10 rows Rl-R10, the columns and rows being formed by lines or markings 43 and 45 on theupper face of the answer card. Each of the spaces in the rows R1 and R10 and columns C1 and C10 is provided with indicia as illustrated in FIG. 10. In row R1 and column C1 the indicia in the various spaces are the consecutive numbers 1-10. In row R10 and column C10 the spaces are marked 10, 20, 30 100. The numbers' in columns C1, C10 and rows R1, R10 provide means for assigning a specific numerical designation to each of the other spaces on the card surface as explained later. Alternatively, member 41 can be a tray somewhat like tray 1 and the numerical indicia can be located on the edges of the tray. This separate tray could fit inside tray 1. The spaces at the intersection of columns C2C9 and rows R2-R9 receive blocks in working multiplication problems. There are no numbers printed in these spaces since numbers in these spaces would be the products of the numbers at the heads of the columns and rows and thus would give the solution to the problem the student is assigned.

FIG. 11 illustrates the coding or pattern indicia appearing on 64 blocks which are used with the answer sheet 41 in FIG. 10. The blocks of FIG. 11 can be 64 of the blocks 5 previously described or they can be 64 separate blocks. When using the blocks 5 previously described the pattern and numerical indicia are on two opposite block faces other than the faces 5a and 5b. In other words, pattern indicia on the blocks shown in FIG. 11 could be on the faces 50 of the blocks while numerical indicia could be on the faces 5d of the blocks. When numerical indicia are provided on two different faces of a single block, then it is preferred that the background of the block faces containing the numerical indicia be colored differently or otherwise distinguished so that the proper pattern face will be face up after the problem has been solved.

When working a multiplication problem, the 64 blocks of FIG. 11 are placed on the 64 spaces formed by the intersection of columns C2-C9 androws R2-R9 on card 41. The numerical indicia on the lower face of each block are the product of the numbers at the head of the column and at the left of the row where it is to be placed. For example, the upper left block is to be placed in column C2, row R2 and it has the number 4 on its lower face. Similarly the block for column C5, row R6 has the number 30 on its lower face. If the numbers in column C1 and row R1 are changed, then the numbers on the blocks will be changed accordingly.

Some numbers are on more than one block of the set of blocks. For example, for the particular multiplication problems illustrated by FIG. 11 the numbers 24, 12 and 18 each appear on four separate blocks and the numbers 16 and 36 appear on three different blocks. The numbers 6, 8, 10, 14, 15, 20, 21, 27, 28, 30, 32, 35, 40, 42, 45, 48, 54,56, 63 and 72 each appears on two of the blocks. The other numbers each appears on only one block. This must be considered in preparing the pattern indicia since the blocks having on their lower face the same numerical indicia (such as the number 24) must have the same pattern on their upper face so that'they are readily interchangeable. The pattern shown in FIG. 11 is prepared so that the pattern indicia on the blocks having the same numbers are identical to each other but distinguishable from all other blocks whereas the blocks having numbers used only once have pattern indicia distinguishable from all other blocks. However, the black border found on all four sides of each of the blocks that form the outside rows and columns of blocks in FIG. 11 form a separate set of blocks and on the numbered side of this set of blocks will be placed an identical black border so that the pupil will know that these blocks must be placed somewhere on the border or perimeter of the answer card.

The pattern indicia generally designated 47 in FIG. 11 comprise various arrangements of background colors and colored bars or diagonal extending lines together with a border. For example, in the upper left in FIG. 11 there is a block which has a solid colored pattern 49 such as orange surrounded by a border 51 of a different color, such as black, for example. The border 51 of the pattern extends around each of the blocks in the outer columns and rows of the set of blocks. Other blocks have a different background color 53 such as light blue, for example. Other background colors are shown at 55, 57 and 59. The color 55 can be yellow or white, for example. The color 57 can be pink and the color 59 can be green.

Pattern 47 includes a plurality of lines or bars which extend diagonally from the upper left to the lower right across the blocks. These lines may be of various colors. For example, the lines designated 61 can be brown, the lines 63 can be red, the lines 65 a dark blue (as distinguished from the light blue background color 53), and the lines 67 can be purple.

In working multiplication problems using the blocks of FIG. ll, all the blocks are placed with the numerical indicia facing up. Then the pupil, using the answer card of FIG. 10, proceeds to work the required multiplication problems by multiplying a number in column C1 by a number in row R1, finding the answer on the numerical indicia on a block, and placing that block (number side down) on the space formed by the intersection of the column and row. For example, the block bearing the indicia 4 is placed number face down on the space where column C2 and row R2, intersect (two times two being four). Similarly, one of the blocks bearing the indicia 30 is placed on the space at the intersection of column C6, row R5 and the other block having indicia 30 is placed on the space at the intersection of column C5, row R6. As will be seen from FIG. 11, these two blocks have exactly the same pattern indicia on their upper face so that they may be interchanged without loss of the readily distinguishable pattern necessary for grading. When all problems have been correctly solved, the pattern on the upper faces of the blocks will appear as shown in FIG. 11. If one or more of the multiplication problems has been incorrectly solved, their the regularity of the pattern will be broken and the incorrectness of the solution is readily determinable by the teacher from an examination of the pattern above.

When learning factoring the pupil picks up the blocks 5 at random one at a time from a container and determines that it is a product of a number in one row and a number in one column, and places the blocks at their intersection. If a block with identical pattern indicia is there then he must select one other row and column which gives the correct answer.

FIG. 12 of the drawings illustrates an answer card similar to the card shown in FIG. 10 but which has a border indicia for teaching fractions or factoring. The blank spaces appearing in columns 2-7 of FIG. 12 and in rows 2-7 will be occupied by a set of blocks somewhat similar to that shown in FIG. 11 and the numerical indicia on one face of the blocks will be the product of the indicia of a particularcolumn and row shown inFIG. 12. For I example, the block which will appear on column C2, row R2 will bear, the indicia 1/44 since c0lumnC2 is marked 1/4 and'row R2 is marked 1/11. Similarly, the

has. a unique indicia or marking such as the pattern shown in FIG. 4. A picture or any other pattern which is easily recognizable as being a unique pattern can be used. In other respects factoring of fractions is the same as the teaching of factoring previously described.

The coding for faces b of the blocks in each of the previously described embodiments has consisted of pattern indicia such as bars, lines and solid colors covering an entire block face. Also, addition and subtraction have been specifically described only in connection with a base ten system. FIGS. 13 and 14 show modified means for coding the blocks and illustrate such in connection with a base four system. FIG. 13 illustrates the Coding or pattern indicia on 16 blocks arranged in four columns C1, C2, C3 and C4 and four rows R1, R2, R3 and R4. In this instance, the face 5b of each block is divided into four triangular segments by line indicia extending diagonally across the rectangular block face. The four triangular segments are designated 71, 73, 75 and 77 for the upper left block, Each of these triangular segments has a base extending the entire length of one side edge of the block and an apex at the center of the block. Each of these face segments may be a different color. For example, the triangular segment 71 may be yellow, segment 73 may be brown, segment 75 may be green and segment 77 may be pink. These various colors are shown by dilferent cross hatching or by areas free of cross hatchings.

Each of the faces 5b of the other blocks in FIG. 13 is divided into four triangular portions which contain various combinations of the four colors. When the blocks are properly arranged to give the correct solution to a problem, any two adjacent edges of any two blocks will have the same colored triangular segments abutting each other as illustrated in FIG. 13. This permits the teacher or instructor to immediately determine whether the problem given has been correctly solved.

In a base four system as shown in FIGS. 13 and 14 the numerical indicia on the answer card (not shown) will designate the spaces occupied by the blocks as follows: In row R1 the spaces will be numbered 1, 2, 3 and in row R2 the spaces will be designated 11, 12, 13, 20; row R3 the spaces will be designated 21, 22, 23 and 30; in row R4 the spaces will be designated 31, 32, 33 and 100. Addition and subtraction problems are solved in the same manner as previously described in connection with FIGS. 8 and 9. The coding on adjacent edges of adjacent blocks always match if the problem has been correctly solved. Since the numerical indicia increases from left-to-right across the rows, the block numbered 1 will move to the right across row R1 as the problems add one, add two, add three are solved, then to the column 1, row R2 position to solve the problem add 10, etc.

In FIG. 14 the coding is obtained by shaping the blocks so that various edges of the faces 5b of the blocks are different. The particular edge shaping illustrated on the upper left block of FIG. 14 comprises concave arcuate edges 79 and 81 and two edges 83 and 85 which have semicircular bosses projecting from the center portion of otherwise straight edges. Edges of other blocks having mating recesses or convex edge portions so that the various blocks fit together somewhat in the manner of a so-called jigsaw puzzle.

Various other types of coding can be used for rapidly grading a problem solution. For example, where only a single solution is to be determined or is to be obtained by solving a specific problem (like in multiplication), then the coding may comprise a picture or design formed by indicia on all of the faces 5b of the blocks when the blocks are properly arranged on an answer card to give the correct solution to the problem. To make such a picture or design for addition and subtraction in any base, a plurality of squares 35 are formed as shown in FIG. 7. A picture or design is then drawn across two of the side edges of the square which are not parallel with each other, e.g., two adjacent side edges. The design on any plurality of squares is then superimposed onto the base square as described in connection with FIG. 7. The center of the base square is then filled in. Each block is compared to be sure no two blocks have pattern indicia which is too similar. Numerical indicia may also be used as the coding on face 5b of the blocks.

Throughout the foregoing description, reference has been to blocks having six distinct faces. This permits a single set of blocks to be used with as many as three different combinations of numerical indicia and coding on opposite faces. However, it will be understood that very thin disks or other shape markers having coding on one surface and numerical indicia on the other surface can also be used.

In the example of blocks shown in the drawings, faces 5a and 5b are used for teaching base 10 addition and subtraction. Teaching of multiplication as shown in FIGS. 10 and 11 would use sides 50 and 5d of 64 blocks, leaving sides 50 and 5d of 36 blocks free to be used for teaching fractions and factoring as shown in FIG. 12. As previously mentioned, a separate tray can be provided containing the FIG. 10 indicia and spaces, and such tray will conveniently. fit within the border 9 of tray 1 (FIG. 1). Similarly, a third tray eight columns wide and eight rows deep can be placed in this second tray, the third tray containing indicia such as shown in FIG. 12. The FIG. 12 data can then be expanded by one row and column. This utilization of the block faces leaves 100 faces 5e and 5f of the blocks available for teaching any other material whose answers can be coded. For example, the numerical value of alphabet letters and their normal sequence can be taught by assigning a value of 1 to the letter A, a value of 2 to the letter B, etc. with Y and Z both on the 25th 'blocks. These blocks would be placed letter side down on a card insert having 100 spaces which had been bisected vertically and horizontally by heavy lines so as to form four groups of spaces each having 25' spaces. Each of the four sets of spaces would be numbered from 1 to 25.

In view of the above, it will be seen that the several objects of the invention are achieved and other advantageous results attained.

As various changes could be made in the above constructions without departing from the scope of the invention, it is intended that all matter contained in the above description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.

What is claimed is:

1. Educational apparatus comprising a surface divided into a plurality of spaces, each space representing a specific number, a plurality of blocks positionable on said spaces, each of the blocks having numerical indicia on a first face, the numerical indicia on each block having a specific mathematical relation to the number which a space on the surface represents, each block containing coding which matches adjacent coding on adjacent blocks when the blocks are arranged on the spaces in such manner that the same mathematical relation exists between the numerical indicia on each of the blocks and the number represented by the space occupied by each block, each block having a rectangular second face, and the coding including a different color along each of the four sides of the second face of each block.

2. Educational apparatus comprising a surface divided into a plurality of spaces, each space representing a specific number, a plurality of blocks positionable on said spaces, each of the blocks having numerical indicia on a first face, the numerical indicia on each block having a specific mathematical relation to the number which a space on the surface represents, each block containing coding which matches adjacent coding on adjacent blocks when the blocks are arranged on the spaces in such manner that the same mathematical relation exists between the numerical indicia on each of the blocks and the number represented by the space occupied by each block, at least two of the blocks having identical numerical indicia and coding, at least one block having numerical indicia and coding different from the indicia on all other blocks, the spaces being arranged in columns and rows, and the number represented by each space being indicated by numerical indicia at the head of each column and row. 3. Educational apparatus comprising a card having a fiat surface divided into a plurality of adjacent spaces, the spaces being arranged in rows and columns and each space representing a specific number,

a plurality of blocks positionable on the spaces on the card, the blocks having faces substantially the same size and shape as the card spaces and being present in sufiicient quantity to substantially cover all of the spaces when the blocks are on the card,

each of the blocks having a first face containing numerical indicia with the numerical indicia on a particular block having a specific mathematical relation to the number represented by a particular space on the card,

each block having a second face containing indicia comprising a portion of a pattern, the pattern indicia on each block being related to the pattern indicia on all other blocks and to the numerical indicia on the blocks so that the pattern indicia on the blocks jointly form an unbroken distinctive recognizable pattern when the blocks are arranged pattern face up on all of the spaces on the card With the numerical indicia on each block having the same mathematical relation to the number represented by the card space occupied by it as the corresponding indicia on all other blocks have to the respective number represented by the card space occupied by them,

the pattern indicia being such that the pattern is broken if at least one block is located on the card so that its numerical indicia does not have the same mathematical relation to the number represented by the space occupied by it as the corresponding indicia on all other blocks have to the respective numbers represented by the spaces occupied by them,

and all of said blocks being positionable on more than one of the spaces of the card to cause a shift in the pattern formed by the indicia on the second faces of the blocks without breaking the pattern.

4. Educational apparatus as set forth in claim 3 wherein the second face of one of the blocks has a distinctive key mark distinguishing said one block from the other 12 blocks whereby changes in the pattern caused by moving all of the blocks can be readily detected.

5. Educational apparatus comprising an answer card having a fiat surface divided into a plurality of adjacent spaces arranged in rows and columns, each space representing a specific number,

a plurality of blocks positionable on the spaces on the answer card, the blocks being substantially the same size and shape as the answer card spaces and being present in suflicient quantity to substantially cover all of the spaces when the blocks are on the answer card,

each of the blocks having a first face containing numerical indicia related to at least one space on the answer card,

each block having a second face containing indicia comprising a portion of a pattern, the pattern indicia on each block being related to the pattern indicia on all other blocks and to the numerical indicia on the blocks so that the pattern indicia on the blocks jointly form an unbroken distinctively recognizable pattern when the blocks are correctly arranged pattern face up on all of the spaces on the answer card with the numerical indicia on each block resting on a related space on the answer card occupied by them,

the pattern indicia being such that the pattern is broken if at least one block is incorrectly located on the answer sheet so that its numerical indicia does not rest on a related space on the answer card,

at least two of said blocks having identical numerical and pattern indicia, and at least one of the blocks having numerical and pattern indicia different from the indicia on all other blocks.

References Cited UNITED STATES PATENTS 898,587 9/1908 Matthias 35-314 2,875,531 3/1959 Mansfield 3573 XR 2,899,756 8/1959 Wise 35-3 1.6 3,212,201 10/1965 Jensen 35-31.4 3,242,594 3/1966 Smith 273157 XR FOREIGN PATENTS 911,228 5/1954 Germany. 744,637 2/ 1956 Great Britain.

EUGENE R. CAPOZIO, Primary Examiner. W. H. GRIEB, Assistant Examiner. 

